In this work, we modify metamorphosis to include a spatially dependent control
parameter \(x\mapsto \nu (x)\)  in order to selectively allow non-diffeomorphic
(metamorphic) matching in parts of the image. For \(\nu (\cdot ) = \sigma ^2\) 
our theory recovers the standard metamorphosis model. However, with a localised
control (e.g. a Gaussian centred at a point in \(\mathbb {R}^{d}\) ), we can selectively
introduce metamorphosis in an image and model local topological effects such as
growth phenomena. The difficulty of this problem is to infer the function
\(\nu (\cdot )\)  without prior knowledge of the location of the topological effects.
This problem is similar to the one treated in [1]},
where such functions were parameterising the randomness in LDDMM matching of
shapes. We will use a Markov chain Monte Carlo (MCMC) approach to infer
appropriate functions \(\nu (\cdot )\) , such that the topological effects are well
described and a large part of the matching remains diffeomorphic. In this paper,
we focus on landmark matching but aim to extend the theory of selective
metamorphosis to data structures amenable to classical metamorphosis or LDDMM
theory can handle.
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